Editing Local zeta function (section)

In [[mathematics]], the '''local zeta function''' {{math|''Z''(''V'',&nbsp;''s'')}} (sometimes called the '''congruent zeta function''' or the [[Hasse–Weil zeta function]]) is defined as

:<math>Z(V, s) = \exp\left(\sum_{k = 1}^\infty \frac{N_k}{k} (q^{-s})^k\right)</math>

where {{mvar|V}} is a [[Singular point of an algebraic variety|non-singular]] {{mvar|n}}-dimensional [[projective algebraic variety]] over the field {{math|'''F'''<sub>''q''</sub>}} with {{mvar|q}} elements and {{math|''N''<sub>''k''</sub>}} is the number of points of {{mvar|''V''}} defined over the finite field extension {{math|'''F'''<sub>''q''<sup>''k''</sup></sub>}} of {{math|'''F'''<sub>''q''</sub>}}.<ref>Section V.2 of {{Citation
| last=Silverman
| first=Joseph H.
| author-link=Joseph H. Silverman
| title=The arithmetic of elliptic curves
| publisher=[[Springer-Verlag]]
| location=New York
| series=[[Graduate Texts in Mathematics]]
| isbn=978-0-387-96203-0
| mr=1329092
| year=1992
| volume=106
}}</ref>

Making the variable transformation {{math|''t''&nbsp;{{=}}&nbsp;''q''<sup>−''s''</sup>,}} gives
:<math>
\mathit{Z} (V,t) = \exp 
\left( \sum_{k=1}^{\infty} N_k \frac{t^k}{k} \right)
</math>
as the [[formal power series]] in the variable <math>t</math>.

Equivalently, the local zeta function is sometimes defined as follows: 
:<math>
(1)\ \ \mathit{Z} (V,0) = 1 \,
</math>
:<math>
(2)\ \ \frac{d}{dt} \log \mathit{Z} (V,t) = \sum_{k=1}^{\infty} N_k t^{k-1}\ .</math>

In other words, the local zeta function {{math|''Z''(''V'',&nbsp;''t'')}} with coefficients in the [[finite field]] {{math|'''F'''<sub>''q''</sub>}} is defined as a function whose [[logarithmic derivative]] generates the number {{math|''N''<sub>''k''</sub>}} of solutions of the equation defining {{mvar|V}} in the degree {{mvar|k}} extension {{math|'''F'''<sub>''q''<sup>''k''</sup></sub>.}}

<!--In [[number theory]], a '''local zeta function'''

:<math>Z(-t)</math>

is a function whose [[logarithmic derivative]] is a [[generating function]]
for the number of solutions of a set of equations defined over a [[finite field]] ''F'', in extension fields ''F<sub>k</sub>'' of ''F''. -->